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242 CHAPTER 12. RSA
Perhaps not surprisingly, given the last several sections, the solution to all
three of these difficulties will come from the same modification: instead of working
modulo 26 we will work modulo P Q, where P and Q are two (large) primes.
From Euler’s generalization of Fermat’s Theorem, we know that a (P −1)(Q−1) ≡ 1
(mod P Q) whenever neither P nor Q divides a. So if we cleverly pick d (using
the Euclidean Algorithm) to be the solution to ed ≡ 1 (mod (P − 1)(Q − 1)),
then we have
(
a
e ) d
≡ a ed ≡ a (mod P Q).
That is, raising to the d-th power will reverse the effect of raising the e-th
power. 9 We will be able to decipher messages.
Further, having two letters that become the same when enciphered will be
impossible: if m e ≡ n e (mod P Q), then
m ≡ m ed ≡ (m e ) d ≡ (n e ) d ≡ n ed ≡ n
(mod P Q).
So two letters (or messages) that are enciphered to the same letter (or message)
were actually the same to start. That is, different letters are enciphered
differently.
Finally, there is the question of choices of e and d: are there enough so that
our supposed enemy cannot stumble upon d simply by trying all the possibilities.
Notice first that different choices of e can lead to the same actual encryption.
We pointed this out in the case of prime moduli: if e ≡ e ′ (mod P − 1) then
m e ≡ m e′ (mod P ). The same is true modulo (P − 1)(Q − 1): if if e ≡ e ′
(mod (P − 1)(Q − 1)) then m e ≡ m e′ (mod P Q). So it doesn’t actually add
more choices to allow e and d to be larger than (P − 1)(Q − 1), and so there
are at most (P − 1)(Q − 1) different choices for e and d. To provide for a large
number of choices for e, then, we will use very large P ’s and Q’s.
12.8 RSA
The RSA 10 crypto-system was invented by Ronald L. Rivest, Adi Shamir and
Leonard Adelman in 1977. We have given the basics of the system. The
only thing left to add is that if we are going to work modulo a large modulus,
there is no need to encipher one letter at a time. We can instead use this as a
polygraphic cipher.
9 This statement is the “trick” behind the cipher system we are about to explain. Make
sure you understand it, looking back at Theorems 12.1 and , if necessary.
10 U.S. Patent No. 4, 405, 829, September 20, 1983, expired on September 20, 2000.